AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 8. DIAGONALIZACION
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
8. DIAGONALIZACION
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
INTRODUCTION
The squared matrix
A =
1 3 3−3 −5 −33 3 1
and the diagonal matrix
D =
1 0 00 −2 00 0 −2
verify
D = P−1AP where P =
1 −1 −1−1 1 01 0 1
.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Let V be a real vector space.
If f : V → V is an endomorphism, whose matrix D in somebasis of V is diagonal, then numerous problems related with fare significantly simplified.
Some examples are: classify f (injective, surjective, bijective);obtain its invariants; compute fn, n ∈ N.
Given a squared matrix A (associated to an endomorphism)we explain next how to obtain a diagonal matrix D related withA by
D = P−1AP where P =
1 −1 −1−1 1 01 0 1
.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
EIGENVALUES AND EIGENVECTORS OF ANENDOMORPHISM
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Let V be a nonzero real vector space and f : V → V an endo-morphism.
Definition A scalar λ ∈ R is an eigenvalue of f if there exists anonzero vector v ∈ V such that
f (v) = λv.
Definition If λ is an eigenvalue of f , a vector v ∈ V verifyingf (v) = λv is called eigenvector of f associated to λ.Example Let f : R3 → R3 be the endomorphism defined by
f (x, y, z) = (x + 3y + 3z,−3x− 5y − 3z, 3x + 3y + z).
Then, λ = −2 is an eigenvalue of f because there exists anonzero vector v = (−1, 1, 0) such that f (v) = −2v. Thus v isan eigenvector of f associated to λ = −2.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Proposition The set of all the eigenvectors of f associated tothe same eigenvalue λ of f
Vλ = {v ∈ V | f (v) = λv}is a vector subspace of V called eigenspace of f associated toλ.Remarks Let id : V → V be the identity map in V .1. Vλ = Ker(f − λid).2. λ ∈ R is an eigenvalue of f ⇔ the endomorphism f −λid is
not one-to-one.
3. In particular, λ = 0 is an eigenvalue of f ⇔ f is not one-to-one⇔ Ker(f ) = V0 6= {0V }.
4. Some endomorphisms do not have eigenvalues, so they donot have eigenvectors either.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Proposition Let λ1, . . . , λp be p different eigenvalues of f .
1. Let vi be a nonzero eigenvector of f associated to λi, i =1, . . . , p then v1, . . . , vp are linearly independent.
2. Vλ1 + · · · + Vλp is a direct sum.
Remark If dimV = n then the endomorphism f will have atmost n different eigenvalues, otherwise it would have morethan n linearly independent eigenvectors.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
EIGENVALUES AND EIGENVECTORS OF A SQUAREDMATRIX
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Let A be a squared n × n matrix whose entries are real num-bers, that is A ∈Mn×n(R).
Definition A scalar λ ∈ R is an eigenvalue of A if there exists anonzero column vector X ∈Mn×1(R) such that
AX = λX.
Definition If λ is an eigenvalue of A, a column vector X ∈Mn×1(R) verifying AX = λX is called an eigenvector of A as-sociated to λ.
Let In be the identity matrix of size n× n.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Remarks Let us assume that dimV = n and let B be a basisof V . Let f : V → V be the endomorphism whose matrix in thebasis B is A. The following statements are verified:
1. λ is an eigenvalue of f ⇔ λ is an eigenvalue of A⇔ det(A−λIn) = 0.
2. Let v ∈ V and let X be the column vector of the coordinatesof v in the basis B, that is X ∈ Mn×1(K). Then, v is aneigenvector of f ⇔X is an eigenvector of A⇔ (A−λIn)X =0.
3. dimVλ = n− rank(A− λIn).
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Example Let f : R3 → R3 be the endomorphism defined by
f (x, y, z) = (3x, x + 2y, 4x + 2z).
Let B be the standard basis of R3. The matrix of f in the basis B is
A =Mf(B) =
3 0 0
1 2 0
4 0 2
.
Since rank(A− 3I3) = 2 the system (A− 3I3)X = 0, that is 3− 3 0 0
1 2− 3 0
4 0 2− 3
x
y
z
=
0
0
0
has a nonzero solution. Therefore, λ = 3 is an eigenvalue of f and A. Theeigenspace V3 has dimV3 = 1 and cartesian equations
Cartesian equations of V3
{x− y = 0
4x− z = 0.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Thus V3 = {(a, a, 4a) | a ∈ R}.On the other hand, rank(A− 2I3) = 1, so λ = 2 is an eigenvalue of f and A,dimV2 = 2. Solving (A− 2I3)X = 0, that is 3− 2 0 0
1 2− 2 0
4 0 2− 2
x
y
z
=
0
0
0
we have V2 = {(0, a, b) | a, b ∈ R} and the cartesian equation of V2 is x = 0.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
ROOTS OF POLYNOMIALS AND THEIR MULTIPLICITY
Let R[x] be the set of all polynomials in x with real coefficients.
Definition Let p(x) be a polynomial in R[x]. A scalar λ ∈ R is aroot of p(x) if p(λ) = 0.Equivalently, λ is a root of p(x) if and only if (x−λ) divides p(x),that is, there exists a polynomial q(x) ∈ R[x] such that
p(x) = (x− λ)q(x).
Definition Let λ be a root of the polynomial p(x). We call multi-plicity of λ to the highest natural number m such that (x − λ)mdivides p(x), so
p(x) = (x− λ)mq(x), q(x) ∈ R[x].
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Every polynomial of degree greater or equal than one has allits roots in C. On the other hand, there exist polynomials in R[x]with no roots in R. As an example, x2 + 1 has real coefficientsbut only complex roots.
Proposition Let p(x) ∈ R[x] be a polynomial of degree n whosereal roots are λ1, . . . , λp with multiplicities m1, . . . ,mp respecti-vely. Then:
1. There exists q(x) ∈ R[x] such that p(x) = (x− λ1)m1 · · · (x−λp)
mpq(x) so m1 + · · · +mp ≤ n.
2. If p(x) ∈ C[x] all its roots are in C so
p(x) = (x− λ1)m1 · · · (x− λp)mp with m1 + · · · +mp = n.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
CHARACTERISTIC POLYNOMIAL
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Definition Let A ∈ Mn×n(R). The characteristic polynomial ofA is det(A− λIn) and its characteristic equation is
det(A− λIn) = 0.
Proposition Let A and A′ be matrices in Mn×n(R). If A andA′ are matrices of the same endomorphism in different basesthen they have the same characteristic polynomial.
Let V be a nonzero real vector space and let f : V → V be anendomorphism.Remark
1. All the matrices associated to f in different bases of V havethe same characteristic polynomial.
2. The converse of the previous proposition is not true.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Definition The characteristic polynomial of f is the characte-ristic polynomial of any of the matrices associated to f in thedifferent bases of V . Analogously with the characteristic equa-tion of f .
Example Let us compute the characteristic polynomial of theendomorphism f of R5 defined by
f (x1, x2, x3, x4, x5) = (−x2, x1 + x3, 2x3 − x4, 2x4 + 6x5, 3x5).
Let B be the standard basis of R5 and A =Mf(B), then:
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
det(A− λI5) =
∣∣∣∣∣∣∣∣∣∣0− λ −1 0 0 01 0− λ 1 0 00 0 2− λ −1 00 0 0 2− λ 60 0 0 0 3− λ
∣∣∣∣∣∣∣∣∣∣=
=(3− λ)(2− λ)2∣∣∣∣ 0− λ −1
1 0− λ
∣∣∣∣=(3− λ)(2− λ)2(λ2 + 1).
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
MULTIPLICITY OF AN EIGENVALUE
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Let V be a nonzero real vector space and f : V → V an endo-morphism. Let A ∈Mn×n(R).
Definition Let λ be an eigenvalue of f (or A). We call multiplicityof λ to its multiplicity as a root of the characteristic equation off (or A).
Theorem Let λ be an eigenvalue of f (or A) with multiplicity m.Then
1 ≤ dimVλ ≤ m.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Remarks
1. Let λ be an eigenvalue of f (or A) with multiplicity m. Ifm = 1 then dimVλ = 1
2. If A,A′ ∈ Mn×n(R) are matrices of the same endomorp-hism in different bases then they have the same eigenva-lues, with the same multiplicities and the same dimensionsof their eigenspaces.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Proposition Let us suppose that dimV = n. Let λ1, . . . λp be thedistinct eigenvalues of f (or A), m1, . . . ,mp their multiplicitiesand d1, . . . , dp the dimensions of the corresponding eigenspa-ces. Then the maximum number of linearly independent eigen-vectors of f (or A) is d1 + · · · + dp. Furthermore,
p ≤ d1 + · · · + dp ≤ m1 + · · · +mp ≤ n.
Remark The characteristic polynomial may not have only realroots and then m1 + · · · +mp < n. In fact, it could have no realroots and therefore no eigenvalues in this case.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Example We obtain next the eigenvalues of the matrix A toget-her with their multiplicities and the dimensions of the eigens-paces,
A =
1 2 102 1 10−1 −1 −6
.
The characteristic polynomial is det(A− λI3) = λ3 + 4λ2 + 5λ +2 = (λ + 2)(λ + 1)2. Then we have eigenvalues λ1 = −2 withmultiplicity m1 = 1 and λ2 = −1 with multiplicity m2 = 2. Thedimensions are
d1 = dimVλ1 = 3− rank(A− λ1I3) = 1,
d2 = dimVλ2 = 3− rank(A− λ2I3) = 2.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
DIAGONALIZATION OF ENDOMORPHISMS ANDSQUARED MATRICES
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Let V be a nonzero real vector space and let f : V → V be anendomorphism. Let A ∈Mn×n(R).
Definition The endomorphism f is diagonalizable if there existsa basis B′ of V so that the matrix Mf(B
′) is diagonal. Then, todiagonalize f is to find B′.
Definition The matrix A is diagonalizable if there exists a dia-gonal matrix D and an invertible matrix P ∈Mn×n(R) such thatD = P−1AP . Then, to diagonalize A means to find D and P .
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Remarks
1. Let us suppose that A is the matrix of f in a basis B. Then:
f is diagonalizable ⇔ A is diagonalizable.
2. If D = P−1AP is a diagonalization of A then
a) D = Mf(B′) is the matrix of f in a basis B′ of V consis-
ting of eigenvectors.b) P = M(B′, B) is the matrix of the change of coordinates
from B′ to B.
Proposition An endomorphism f is diagonalizable if and only ifthere exists a basis of V consisting of eigenvectors of f .
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Theorem Let us suppose that dimV = n. Let λ1, . . . , λp be thedistinct eigenvalues of f (or A), m1, . . . ,mp their multiplicitiesand d1, . . . , dp the dimensions of the eigenspaces. The followingare necessary and sufficient conditions for the existence of abasis of V consisting of eigenvectors.
1. The characteristic polynomial of f has only real roots
λ1, . . . , λp
, that ism1 + . . . +mp = n.
2. The multiplicity of each eigenvalue equals the dimension ofits eigenspace, this is
mi = di, i = 1, . . . , p.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Corollary Let us suppose that f is diagonalizable. Let λ1, . . . , λpbe the eigenvalues of f with multiplicities m1, . . . ,mp respecti-vely. Let Bi be a basis of the eigenspace Vλi having mi = dielements, i = 1, . . . , p. Then:
1. B′ = B1 ∪ · · · ∪Bp is a basis of V consisting of eigenvectorsof f .
2. The matrix of f in the basis B′ is diagonal and its maindiagonal contains the elements
λ1, m1. . ., λ1, λ2, m2. . ., λ2, . . . , λp,mp. . ., λp
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Examples Let f : R3 → R3 be the endomorphism defined by
f (x, y, z) = (x + 2y + 10z, 2x + y + 10z,−x− y − 6z)
whose matrix in the standard basis B of R3 is a matrix A whoseeigenvalues are λ1 = −2 and λ2 = −1. A basis of Vλ1 is
Bλ1 = {(−2,−2, 1)}
and of Vλ2 isBλ2 = {(−5, 0, 1), (−1, 1, 0)}.
Then a basis of R3 consisting of eigenvectors of f is
B′ = Bλ1 ∪Bλ2 = {(−2,−2, 1), (−5, 0, 1), (−1, 1, 0)}.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Finally,
D = P−1AP =Mf(B′) =
−2 0 00 −1 00 0 −1
where
P =
−2 −5 −1−2 0 11 1 0
.